Is there a characterization of injective $C(X)$-modules analogous to Serre-Swan?

The Serre - Swan theory in geography claims that if $X$ is portable Hausdorff and also $C(X)$ the ring of continual features on $X$, after that the group of finitely created projective $C(X)$ - components amounts the group of vector bundles over $X$. Exists a similar theory for the twin idea of injective components?

2022-06-07 14:34:17
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This is not actually a solution. Instead, it manages a various inquiry, particularly the analogue of your inquiry in algebraic geometry. Probably it is still of some passion.

In algebraic geometry, one has an analogue of Serre - - Swan, in which f.g. projective components over a ring (commutative, with 1) $A$ represent limited ranking in your area free sheaves on Spec $A$.

If I remember appropriately, under some presumptions, one can get injective $A$ - components as neighborhood cohomology sheaves of the framework sheaf sustained at (not always shut) factors of Spec $A$.

E.g. if $A = k[x]$ (with $k$ an algebraically shut area), after that the area of sensible features $k(x)$ is gotten as the neighborhood cohomology sheaf $\mathcal H^0_{\eta}(\mathcal O)$, where $\eta$ is the common factor of Spec $A$, while for a shut factor $(x-a)$, the injective component $k[x,1/(x-a)]/k[x]$ is gotten as the neighborhood cohomology sheaf $\mathcal H^1_{(x-a)}(\mathcal O).$

This building and construction results from Grothendieck, I assume, and also is reviewed in Hartshorne is publication Residues and also duality , as a means of creating approved injective resolutions.

E.g. when $A = k[x]$, we have the injective resolution $$0 \to k[x] \rightarrow k(x) \to \bigoplus_{a \in k} k[x,1/(x-a)]/k[x] \to 0,$$ which can be revised in regards to sheaves on $X = $ Spec $A$ as $$0 \to \mathcal O_X \to \mathcal H^0_{\eta}(X,\mathcal O_X) \to \bigoplus_{a \in k} \mathcal H^1_{(x-a)}(X,\mathcal O_X) \to 0;$$ so the framework sheaf has an injective resolution whose $p$th term entails an amount over neighborhood cohomology sheaves sustained at codimension $p$ factors.

I do not have much of a sensation regarding whether one can lug this over in any kind of valuable means to the topological setup, given that the geometry of shut parts in algebraic geometry is far more inflexible than as a whole topology.

2022-06-07 15:01:51